login
This site is supported by donations to The OEIS Foundation.

 

Logo

Invitation: celebrating 50 years of OEIS, 250000 sequences, and Sloane's 75th, there will be a conference at DIMACS, Rutgers, Oct 9-10 2014.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A069283 a(n) = -1 + number of odd divisors of n. 8
0, 0, 0, 1, 0, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 3, 0, 1, 2, 1, 1, 3, 1, 1, 1, 2, 1, 3, 1, 1, 3, 1, 0, 3, 1, 3, 2, 1, 1, 3, 1, 1, 3, 1, 1, 5, 1, 1, 1, 2, 2, 3, 1, 1, 3, 3, 1, 3, 1, 1, 3, 1, 1, 5, 0, 3, 3, 1, 1, 3, 3, 1, 2, 1, 1, 5, 1, 3, 3, 1, 1, 4, 1, 1, 3, 3, 1, 3, 1, 1, 5, 3, 1, 3, 1, 3, 1, 1, 2, 5, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,10

COMMENTS

Number of nontrivial ways to write n as sum of at least 2 consecutive integers. That is, we are not counting the trivial solution n=n. E.g., a(9)=2 because 9 = 4 + 5 and 9 = 2 + 3 + 4. a(8)=0 because there are no integers m and k such that m + (m+1) + ... + (m+k-1) = 8 apart from k=1, m=8. - Alfred Heiligenbrunner (alfred.heiligenbrunner(AT)gmx.at), Jun 07 2004

Also number of sums of sequences of consecutive positive integers excluding sequences of length 1 (e.g., 9 = 2+3+4 or 4+5 so a(9)=2). (Useful for cribbage players.) - Michael Gilleland, Dec 29 2002

Let M be any positive integer. Then a(n) = number of proper divisors of M^n + 1 of the form M^k + 1.

This sequence gives the distinct differences of triangular numbers Ti giving n : n = Ti - Tj; none if n = 2^k. If factor a = n or a > (n/a - 1)/2 : i = n/a + (a - 1)/2; j = n/a - (a+1)/2. Else : i = n/2a + (2a - 1)/2; j = n/2a - (2a - 1)/2. Examples: 7 is prime; 7 = T4 - T2 = (1 + 2 + 3 + 4) - (1 + 2) (a = 7; n/a = 1). The odd factors of 35 are 35, 7 and 5; 35 = T18 - T16 (a = 35) = T8 - T1 (a = 7) = T5 - T7 (a = 5). 144 = T20 - T11 (a = 9) = T49 - T46 (a = 3). - M. Dauchez (mdzzdm(AT)yahoo.fr), Oct 31 2005

Also number of partitions of n into the form 1 + 2 + ...( k - 1) + k + k + ... + k for some k >= 2. Example: a(9) = 2 because we have [2, 2, 2, 2, 1] and [3, 3, 2, 1]. - Emeric Deutsch, Mar 04 2006

a(n) is the number of non-trivial runsum representations of n, and is also known as the politeness of n. - Ant King, Nov 20 2010

REFERENCES

Graham, Knuth and Patashnik, Concrete Mathematics, 2nd ed. (Addison-Wesley, 1994), see exercise 2.30 on p. 65.

Apostol, Tom M.; Sums of Consecutive Positive Integers, The Mathematical Gazette, Vol. 87, No. 508, (March 2003), pp. 98-101.

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

A. Heiligenbrunner, Sum of adjacent numbers (in German).

Wikipedia, Polite Number

FORMULA

a(n) = 0 if and only if n = 2^k.

a(n) = 1 if and only if n = 2^k x p where k >= 0 and p is an odd prime. - Ant King, Nov 20 2010

G.f.: sum(k>=2, x^(k(k + 1)/2)/(1 - x^k) ). - Emeric Deutsch, Mar 04 2006

If n = 2^k p1^b1 p2^b2 ... pr^br, then a(n) = (1 + b1)(1 + b2) ... (1 + br) - 1. - Ant King, Nov 20 2010

EXAMPLE

a(14) = 1 because the divisors of 14 are 1, 2, 7, 14, and of these, two are odd, 1 and 7, and -1 + 2 = 1.

a(15) = 3 because the divisors of 15 are 1, 3, 5, 15, and of these, all four are odd, and -1 + 4 = 3.

a(16) = 0 because 16 has only one odd divisor, and -1 + 1 = 0.

Using Ant King's formula: a(90) = 5 as 90 = 2^1 * 3^2 * 5^1, so a(90) = (1 + 2) * (1 + 1) - 1 = 5. - Giovanni Ciriani, Jan 12 2013

x^3 + x^5 + x^6 + x^7 + 2*x^9 + x^10 + x^11 + x^12 + x^13 + x^14 + ...

MAPLE

g:=sum(x^(k*(k+1)/2)/(1-x^k), k=2..20): gser:=series(g, x=0, 115): seq(coeff(gser, x, n), n=0..100); # Emeric Deutsch, Mar 04 2006

MATHEMATICA

g[n_] := Module[{dL = Divisors[2n], dP}, dP = Transpose[{dL, 2n/dL}]; Select[dP, ((1 < #[[1]] < #[[2]]) && (Mod[ #[[1]] - #[[2]], 2] == 1)) &] ]; Table[Length[g[n]], {n, 1, 100}]

Table[Length[Select[Divisors[k], OddQ[#] &]] - 1, {k, 100}] (* Ant King, Nov 20 2010 *)

PROG

(Haskell)

a069283 0 = 0

a069283 n = length $ tail $ a182469_row n

-- Reinhard Zumkeller, May 01 2012

{a(n) = if( n<1, 0, sumdiv( n, d, d%2) - 1)} /* Michael Somos, Aug 07 2013 */

CROSSREFS

Cf. A001227, A062397, A057934. Equals A001227(n) - 1. A138591.

Cf. A182469.

Sequence in context: A046214 A232088 A115413 * A033630 A220122 A101446

Adjacent sequences:  A069280 A069281 A069282 * A069284 A069285 A069286

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, Mar 13 2002

EXTENSIONS

Edited by Vladeta Jovovic, Mar 25 2002

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified August 30 07:15 EDT 2014. Contains 246216 sequences.